The order of a linearly invariant family in Cn

We study the (trace) order of the linearly invariant family in the ball Bn defined by ‖SF‖≤α, where F:Bn→Cn is locally biholomorphic and SF is the Schwarzian operator. By adapting Pommerenke's approach, we establish a characteristic equation for the extremal mapping that yields an upper bound for the order of the family in terms of α and the dimension n. Lower bounds for the order are established in similar terms by means of examples.